Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $z \neq 0$. $q = \dfrac{-7z + 70}{8z^2 - 24z} \div \dfrac{z - 9}{z^2 - 12z + 27} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{-7z + 70}{8z^2 - 24z} \times \dfrac{z^2 - 12z + 27}{z - 9} $ First factor the quadratic. $q = \dfrac{-7z + 70}{8z^2 - 24z} \times \dfrac{(z - 3)(z - 9)}{z - 9} $ Then factor out any other terms. $q = \dfrac{-7(z - 10)}{8z(z - 3)} \times \dfrac{(z - 3)(z - 9)}{z - 9} $ Then multiply the two numerators and multiply the two denominators. $q = \dfrac{ -7(z - 10) \times (z - 3)(z - 9) } { 8z(z - 3) \times (z - 9) } $ $q = \dfrac{ -7(z - 10)(z - 3)(z - 9)}{ 8z(z - 3)(z - 9)} $ Notice that $(z - 9)$ and $(z - 3)$ appear in both the numerator and denominator so we can cancel them. $q = \dfrac{ -7(z - 10)\cancel{(z - 3)}(z - 9)}{ 8z\cancel{(z - 3)}(z - 9)} $ We are dividing by $z - 3$ , so $z - 3 \neq 0$ Therefore, $z \neq 3$ $q = \dfrac{ -7(z - 10)\cancel{(z - 3)}\cancel{(z - 9)}}{ 8z\cancel{(z - 3)}\cancel{(z - 9)}} $ We are dividing by $z - 9$ , so $z - 9 \neq 0$ Therefore, $z \neq 9$ $q = \dfrac{-7(z - 10)}{8z} ; \space z \neq 3 ; \space z \neq 9 $